Tensor Factor Model Estimation by Iterative Projection

Abstract

Tensor time series, which is a time series consisting of tensorial observations, has become ubiquitous. It typically exhibits high dimensionality. One approach for dimension reduction is to use a factor model structure, in a form similar to Tucker tensor decomposition, except that the time dimension is treated as a dynamic process with a time dependent structure. In this paper we introduce two approaches to estimate such a tensor factor model by using iterative orthogonal projections of the original tensor time series. These approaches extend the existing estimation procedures and improve the estimation accuracy and convergence rate significantly as proven in our theoretical investigation. Our algorithms are similar to the higher order orthogonal projection method for tensor decomposition, but with significant differences due to the need to unfold tensors in the iterations and the use of autocorrelation. Consequently, our analysis is significantly different from the existing ones. Computational and statistical lower bounds are derived to prove the optimality of the sample size requirement and convergence rate for the proposed methods. Simulation study is conducted to further illustrate the statistical properties of these estimators.

Figures (9)

• Figure 1: Experiment 1 under Configuration I. Boxplot of the logarithm of the estimation error of $A_1$. Nine methods (i)-(ix) are considered in total. Three rows correspond to three signal-to-noise strengths $\lambda=1,2,4$. Two columns correspond to two sample sizes $T=256,1024$.
• Figure 2: Experiment 1 under Configuration I. Trajectory of the logarithm of the estimation error of $A_1$ with fixed sample size $T=256$. Two rows correspond to two signal-to-noise strengths $\lambda=1,2$. Two columns correspond to TIPUP-based and TOPUP-based methods respectively.
• Figure 3: Experiment 2 under Configuration I. Boxplot of the logarithm of the estimation error of $A_1$. Six methods (iv)-(ix) with five choices of $h_0$ are considered in total. Two rows correspond to two signal-to-noise strengths $\lambda=2,4$. Two columns correspond to two sample sizes $T=256,1024$.
• Figure 4: Experiment 3 under Configuration I. Boxplot of the estimation error of $A_1$. Six methods (iv)-(ix) are considered in total. Four rows correspond to four signal-to-noise strengths $\lambda=1,2,4,8$. Five columns correspond to five sample sizes $T=16, 64, 256, 1024, 4096$. This figure corroborates the theoretical lower bounds on the sample size in \ref{['eq:sample:itopup']} and \ref{['eq:sample:itipup']}.
• Figure 5: Experiment 1 under Configuration II. Boxplot of the logarithm of the estimation error of $A_1$. Eight methods are considered in total. Two rows correspond to two sample sizes $T=512,1024$. Two columns correspond to two choices of $h_0$. The population signal strengths $\lambda_1^2$\ref{['lam_k']} and $\lambda_1^{*2}$\ref{['lam^*_k']} for different $h_0$ are provided on the top.

Theorems & Definitions (46)

• Remark 2.1
• Remark 2.2: Rank determination
• Proposition 3.1
• Remark 3.1
• Theorem 3.1
• Remark 3.2
• Remark 3.3
• Remark 3.4
• Remark 3.5
• Theorem 3.2